Research Article  Open Access
A Discontinuous Finite Volume Method for the DarcyStokes Equations
Abstract
This paper proposes a discontinuous finite volume method for the DarcyStokes equations. An optimal error estimate for the approximation of velocity is obtained in a meshdependent norm. Firstorder error estimates are derived for the approximations of both velocity and pressure. Some numerical examples verifying the theoretical predictions are presented.
1. Introduction
The study of discontinuous Galerkin methods has been a very active research field since they were proposed by Reed and Hill [1] in 1973. Discontinuous Galerkin methods use discontinuous functions as finite element approximation and enforce the connections of the approximate solutions between elements by adding some penalty terms. The flexibility of discontinuous functions gives discontinuous Galerkin methods many advantages, such as high parallelizability and localizability. Arnold et al. [2] provided a framework for the analysis of a large class of discontinuous Galerkin methods for secondorder elliptic problems.
Based on the advantages of using discontinuous functions for approximation in discontinuous Galerkin methods, it is natural to consider using discontinuous functions as trial functions in the finite volume method, which is called the discontinuous finite volume method. Such a method has the flexibility of the discontinuous Galerkin method and the simplicity and conservative properties of the finite volume method. Ye [3] developed a new discontinuous finite volume method and analyzed it for the secondorder elliptic problem. Bi and Geng [4] proposed the semidiscrete and the backward Euler fully discrete discontinuous finite volume element methods for the secondorder parabolic problems. Ye [5] considered the discontinuous finite volume method for solving the Stokes problems on both triangular and rectangular meshes and derived an optimal order error estimate for the approximation of velocity in a meshdependent norm and firstorder error estimates for the approximations of both velocity and pressure.
The DarcyStokes problem is interesting for a variety of reasons. Apart from being a modeling tool in its own right, it also appears, less obviously, in timestepping methods for Stokes and for high Reynolds number flows (where of course the convective term causes additional difficulties). In [6], the nonconforming CrouzeixRaviart element is stabilized for the DarcyStokes problem with terms motivated by a discontinuous Galerkin approach. In [7], a new stabilized mixed finite element method is presented for the DarcyStokes equations.
In this paper, we will extend the discontinuous finite volume methods to solve the DarcyStokes equations. In our methods, velocity is approximated by discontinuous piecewise linear functions on triangular meshes and by discontinuous piecewise rotated bilinear functions on rectangular meshes. Piecewise constant functions are used as the test functions for velocity in the discontinuous finite volume methods. We obtained an optimal error estimate for the approximation of velocity in a meshdependent norm. Firstorder error estimates are derived for the approximations of both velocity and pressure. For the sake of simplicity and easy presentation of the main ideas of our method, we restrict ourselves to the model problem.
We consider the DarcyStokes equationswhere is a bounded polygonal domain in with boundary . is the velocity, is the pressure, and is a given force term. We assume .
2. Discontinuous Finite Volume Formulation
Let be a triangular or rectangular partition of . The triangles or rectangles in are divided into three or four subtriangles by connecting the barycenter of the triangle or the center of the rectangles to their corner nodes, respectively. Then we define the dual partition of the primal partition to be the union of the triangles shown in Figures 1 and 2 for both triangular and rectangular meshes.
Let consist of all the polynomials with degree less than or equal to defined on . We define the finite dimensional trial function space for velocity on a triangular partition by and on rectangular partition by where denotes the space of functions of the form on .
Let be the finite dimensional space for pressure where Define the finite dimensional test function space for velocity associated with the dual partition as
Multiplying (1.1a) and (1.1b) by and , respectively, we have where is the unit outward normal vector on .
Let be the triangles in , where for triangular meshes and for rectangular meshes, as shown as Figures 3 and 4. Then we have where .
For vectors and , let denote the matrix whose th component is as in [5]. For two matrix valued variables and , we define . Let . Let be an interior edge shared by two elements and in . We define the average and jump on for scalar , vector , and matrix , respectively. If , where and are unit normal vectors on e pointing exterior to and , respectively. We also define a matrix valued jump for a vector as If , define A straightforward computation gives Let . Using (2.7), (2.12), and the fact that for on , (2.7) becomes Since for on , we also have Let . Define a mapping , where is the length of the edge .
We define two norms for as follows: where , ,, and diameter of .
As in [5], the standard inverse inequality implies that there is a constant such that
Lemma 2.1. There exists a positive constant independent of such that
Proof. As in [4], where . Since , we have . Note that is a piecewise linear function, and . By Lemma 3.6 in [4], , we have .
Lemma 2.2 (see [4]). The operator is selfadjoint with respect to the inner product, . Define . Then and are equivalent; here the equivalence constants are independent of . And .
Let It is clear that the solutions of the DarcyStokes equations (1.1a)–(1.1c) satisfy the following: Define the following bilinear forms: Then systems (2.21) are equivalent to
We propose two discontinuous finite volume formulations based on modification of the weak formulation (2.23) for DarcyStokes problem (1.1a)–(1.1c). Let us introduce the bilinear forms as follows: where is a parameter to be determined later. For the exact solution of (1.1a)–(1.1c), we have Therefore, it follows from (2.23) that The corresponding discontinuous finite volume scheme seeks , such that
Let be an edge of element . It is well known (see [2]) that there exists a constant such that for any function , where depends only on the minimum angle of .
Let and be the functions whose restriction to each element is equal to and , respectively.
Lemma 2.3. For , there exists a positive constant independent of such that
Proof. Let , By Lemma 3.1 in [5],
Lemma 2.4 (see [5]). For any , one has
Lemma 2.5 (see [5]). For , there exists a positive constant independent of such that If , then
Lemma 2.6. For any , there is a constant independent of such that for large enough
Proof. Using the proof of Lemmas 3.1 and 3.5 in [5], for , we have when is large enough.
The value of depends on the constant in the inverse inequality. Therefore, the value of for which is coercive is mesh dependent. We introduce a second discontinuous finite volume scheme which is parameter insensitive. Define a bilinear form as follows: Similar to the bilinear form , for the exact solution of the DarcyStokes problem we have Consequently, the solution of the DarcyStokes problem satisfies the following variational equations: Our second discontinuous finite volume scheme for (1.1a)–(1.1c) seeks , such that
For any value of , we have Similarly, we can prove that Let or . In the rest of the paper, we assume that the following is true: If , (2.45) holds for any . If , (2.45) holds for only large enough.
3. Error Estimates
We will derive optimal error estimates for velocity in the norm and for pressure in the norm. A firstorder error estimate for velocity in norm will be obtained.
Let be an interior edge shared by two elements and in . If , we say that is continuous on . We say that is zero at if . Define a subspace of by for rectangular meshes and by for triangular mesh.
It has been proven in [8, 9] that the following discrete infsup condition is satisfied; that is, there exists a positive constant such that
Lemma 3.1. The bilinear form satisfies the discrete infsup condition where is a positive constant independent of the mesh size .
Proof. For and , we have , and . By PoincareFriedrichs , with (3.3), and (2.17) we get for any With , we have proven (3.4).
Define an operator or . For all , where , are the sides of the element . if is a triangle and if is a rectangle. It was proven in [8] that For all , define by Using the definition of and integration by parts, we can show that The CauchySchwarz inequality implies Equations (2.28) and (3.8) imply that The definitions of the norm , (3.7), and (3.11) give
Theorem 3.2. Let be the solution of (2.27), and let ) be the solution of (1.1a)–(1.1c). Then there exists a constant independent of such that
Proof. Let , where is projection from . Then . Subtracting (2.26) from (2.27) and using Lemma 2.4, we get error equationsBy letting in (3.15a) and in (3.15b), the sum of (3.15a) and (3.15b) gives
Thus, it follows from the coercivity (2.45), the boundedness (2.30), (2.44), and (2.34) that
which implies the following:
The previous estimate can be rewritten as
Now using the triangle inequality, (3.7), the definition of , and the inequality mentioned previously, we get
which completes the estimate for the velocity approximation.
Discrete infsup condition (3.4), (3.15a), (3.15b), Lemmas 2.5, 2.4, and inverse inequality give
Using the previous inequality and the triangle inequality, we have completed the proof of (3.13).
Using Lemma 2.1, (3.12), and (3.13), we have
Equations (3.22) and (3.7) and the triangle inequality imply (3.14). We have completed the proof.
4. Numerical Experiments
In this section, we present a numerical example for solving the problems (1.1a)–(1.1c) by using the discontinuous finite volume element method presented with (2.27) and (2.42). Let , be the Delaunay triangulation generated by EasyMesh [10] over with mesh size as shown in Figure 5. We consider the case of , the exact velocity , and the pressure . Denote the numerical solution as and with step which is used to generate the mesh data in the EasyMesh input file, and . For , the numerical results are presented in Tables 1 and 2. It is observed from the tables that the numerical results support our theory.


Acknowledgments
This paper is supported by the Excellent Young and MiddleAged Scientists Research Fund of Shandong Province (2008BS09026), National Natural Science Foundation of China (11171193), and National Natural Science Foundation of Shandong Province (ZR2011AM016).
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Copyright
Copyright © 2012 Zhe Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.